MEN OF MATHEMATICS
some years before Kronecker's death in 1891. Could Canto?
have lived till to-day he might have taken a just pride in the
movement toward more rigorous thinking in all mathematics
for which his own efforts to found analysis (and the infinite) on
a sound basis were largely responsible.
Looking back over the long struggle to make the concepts of
real number^ continuity, limit, and infinity precise and consis-
tently usable in mathematics, we see that Zeno and Eudoxus
were not so far in time from Weierstrass, Dedekind, and Cantor
as the twenty-four or twenty-five centuries which separate
modern Germany from ancient Greece might seem to imply.
There is no doubt that we have a clearer conception of the
• nature of the difficulties involved than our predecessors had,
because we see the same unsolved problems cropping up in new "
guises and in fields the ancients never dreamed of, but to say
that we have disposed of those hoary old difficulties is a gross
mis-statement of fact. Nevertheless the net score records a
greater gain than any which our predecessors could rightfully
claim. We are going deeper than they ever imagined necessary,
and we are discovering that some of the 'laws5 - for instance
those of Aristotelian logic - which they accepted iix theii
reasoning are better replaced by others - pure conventions -
in our attempts to correlate our experiences. As has already
been said, Cantor's revolutionary work gave our present acti-
vity its initial impulse. But it was soon discovered - twenty-one
years before Cantor's death - that his revolution was either too
revolutionary or not revolutionary enough. The latter now
appears to be the case.
The first shot in the counter-revolution was fired in 1897 by
the Italian mathematician Burali-Forti who produced a flagrant
contradiction by reasoning of the type used by Cantor in his
theory of infinite sets- This particular paradox was only the
first of several, and as it would require lengthy explanations to
make it intelligible, we shall state instead Russell's of 1908.
We have already mentioned Frege, who gave the 'class of all
classes similar to a given class' definition of the cardinal number
of the given class. Frege had spent years trying to put the
mathematics of numbers on a sound logical basis. His life work